William W. answered • 09/24/19
Top Pre-Calc Tutor
This is an exponential problem because interest is added on top of interest so it grows much faster than just a linear type equation.
An exponential equation has the generic form y = Vi*(1 +/- r)t where y is some value in the future, Vi is the initial value, r is the rate of change (in decimal or fraction form), and t is time.
First of all, let me explain the +/- in the equation. Sometimes exponential equations grow (like population growth or savings accounts. For those, we use a "+". Sometimes exponential equations decay (go down) like in the depreciation of the value of a car. For those we use "-". You have to choose which one it is, growth or decay, and use the correct one. You don't use them both at once. In this case the value is growing from the initial amount (which we don't know) to the final amount, $20,000 so we'll use a "+".
The critical part of this generic equation though is that time (t) needs to be in the same units as the rate of change (r). For instance, in this case, the problem says the interest (growth rate) is compounded monthly. That means we need to consider t in months but the interest is given as a rate per year. These don't match. So we'll convert both to monthly values. We take t = 3 years x 12 months/yr = 36 months and we'll take the interest of 4%, convert to decimal, 0.04/year x 1yr/12 months = .04/12 = 0.003333 per month. [Please note that books will show a special form of the generic equation for this that looks like this: y = VI(1 + r/n)nt where n is the number of compounding periods. That is exactly what we did and you don't really need a different equation, you just need to make sure the units match.]
In this case, they are looking for the present value which is the same as what I called initial value or Vi. So we can plug in the values given like this:
20000 = Vi(1 + 0.003333)36 or Vi = 20000/(1.003333)36 = $17,741.95
